Second, recall from section REF  that a half-integrable para-Hermitian structure defines
a 2D twistor space, say \(\mathcal {T}_{\beta }\) .
In [1]}, functions \(\varepsilon \)  satisfying (REF ) were interpreted as
defining sections of line bundles over \(\mathcal {T}_{\beta }\) .
Here we see that (REF ) appear as integrability conditions for deformations of a para-complex structure in \(M\) .
We can attempt a more or less “intuitive” understanding of this by considering a picture similar to
figure REF , in which 2D twistor spaces are represented as vertical lines.
We illustrate this in figure REF .
If \(\mathcal {L}\)  is a line bundle over \(\mathcal {T}_{\beta }\) , a section is a smooth map
\(\mathcal {T}_{\beta }\rightarrow \mathcal {L}\) , defined by the condition that it should be covariantly constant over
the twistor surfaces associated to \(\mathcal {T}_{\beta }\) , i.e. we can think of it as a scalar field \(\varepsilon \)  on \(M\) 
such that \(\mathcal {C}_{A^{\prime }}\varepsilon =0\) .
So a section of \(\mathcal {L}\)  defines, by theorem REF , a half-integrable small deformation,
and consequently a 2D twistor space \(\mathcal {T}_{\beta _{\varepsilon }}\) , “close to” \(\mathcal {T}_{\beta }\) ,
see fig. REF .
